Carlos Esteve-Yagüe

Olivier Bokanowski, Carlos Esteve-Yagüe, Richard Tsai

We establish a well-posedness and error-estimation framework that solves Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations. This approach also applies naturally to second-order elliptic and parabolic problems. We prove that, under suitable monotonicity conditions, every critical point of the residual loss functional is the unique global minimizer and coincides with the solution of the discrete scheme. We derive \emph{a~posteriori} error estimates that bound the approximation error by the magnitude of the residual with explicit, computable constants, and extend the full analysis to time-dependent problems with implicit discretization of the time derivatives. A spectral analysis of the linearized system shows that the condition number scales as $O(Δx^{-1})$ for proper schemes, and as $O(\exp(Δx^{-1}))$ under a uniform ellipticity condition. These results quantify the increasing difficulty of solving the optimization problem on finer meshes, and motivates a progressive multi-level warm-start strategy using Artificial Neural Networks. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee that the solutions obtained converge to the unique viscosity solution as the mesh is refined. Numerical experiments demonstrate the scalability of the approach to high-dimensional Eikonal equations, level-set problems, and Hamilton--Jacobi--Isaacs equations with genuine second-order diffusion arising from stochastic differential games.

Carlos Esteve-Yagüe, Borjan Geshkovski

We consider the neural ODE and optimal control perspective of supervised learning, with $\ell^1$-control penalties, where rather than only minimizing a final cost (the \emph{empirical risk}) for the state, we integrate this cost over the entire time horizon. We prove that any optimal control (for this cost) vanishes beyond some positive stopping time. When seen in the discrete-time context, this result entails an \emph{ordered} sparsity pattern for the parameters of the associated residual neural network: ordered in the sense that these parameters are all $0$ beyond a certain layer. Furthermore, we provide a polynomial stability estimate for the empirical risk with respect to the time horizon. This can be seen as a \emph{turnpike property}, for nonsmooth dynamics and functionals with $\ell^1$-penalties, and without any smallness assumptions on the data, both of which are new in the literature.